How much difference does an anti-vibration audio rack make

Good racks matter. A lot. I was skeptical and resistant on these for the past 10-12 years, until I put them in my own system.

And they don’t have to be Uber expensive. Something like my Mapleshade Samson rack will do well. Not pretty, but it does the job well.

The rack you are considering looks fabulous.  It appears to have some serious engineering behind it.  I say go for it.  Start with your preamp and source component.  If I understand the modular design aspect, you can expand to include other parts of your system at a later point.
I cannot answer whether the Aeon will sound truly better than your current wood furniture.  It's the type of product that you have to try and see what it does.  You should have an in-depth conversation with your dealer about how you can accomplish this.
Please post what you do.

You want a solid, stable rack to capably support your gear.  Each piece of kit should have a proper footer underneath it to manage/eliminate vibrations getting to your gear.

One of the most effective and reasonably priced footers can be obtained from Herbie's Audio Lab.  I use their "Tenderfeet" under each piece of my system to great effect.  

Another advantage to your system is decoupling your speakers/subwoofers from the floor.  Allowing the vibrational energy of the speakers to dissipate across the floor rather than be vibrated into your rack will do wonders.  BTW... Herbie's has decouplers too.

I have all my equipment resting on the "holes" that were cut from solid blocks of MIL SPEC rubber, about 1-1/2" thick that are used to support pipes in pipe hangers on SSN688 class submarines......the hanger shop would just throw the "holes" in a trash bin and I would now and then go and get a few......nothing illegal about it. I just happened to be a liaison between manufacturing and engineering at Newport News Shipbuilding back in the 80’s. Also, my floor is concrete, so I have pretty solid and damped support. Components sit in a solid wood frame with glass shelves. No complaints.

Ironically perhaps and certainly counterintuitively solid rigid racks are not really good for the sound, especially if one assumes that solid rigid racks provide any isolation against footfall and other seismic vibration. On the other hand real isolation devices can be mounted on shelves of solid and rigid racks. The Flexy Rack of yore was the opposite concept of the rigid rack inasmuch as it capitalized on the fact that the best isolation is when movement is easier in a particular direction when impelled by an external force, I.e., seismic vibration. Since rigid racks do not allow ease of motion in any direction they have little if any value for vibration isolation. In fact they tend to magnify the effects of vibration. Like tall buildings. They look good, though.